Problem 40 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Sketch the region of integration for the given integral and set up an equivalent integral with the order of integration reversed.\(\int_{0}^{4} \int_{y / 2}^{\sqrt{y}} f(x, y) d x d y\)

Step-by-Step Solution

Answer

The reversed integral is \(\frac{y}{2} \leq x \leq 2 \) for y = x^2 to y = 2x.

Step 1 - Identify the Region of Integration

Examine the given limits of the integral. For the inner integral \(\frac{y}{2} \leq x \leq \sqrt{y} \) and for the outer integral \(0 \leq y \leq 4\). This means x ranges from \(\frac{y}{2}\) to \( \sqrt{y}\) as y varies from 0 to 4.

Step 2 - Sketch the Region

Sketch the curves \(x = \frac{y}{2}\) and \(x = \sqrt{y}\). Note where they intersect to determine the limits for y and x. The intersection points can be found by solving \(\frac{y}{2} = \sqrt{y}\). Solving this gives y = 0 and y = 4. Draw this on the xy-plane.

Step 3 - Identify the New Limits of Integration

To switch the order of integration, observe the region from the perspective of x. The region bounded by \(\frac{y}{2} \leq x \leq \sqrt{y}\) when 0 \leq y \leq 4 can be described as being bounded by \(0 \leq x \leq 2\) and corresponding y limits.

Step 4 - Set Up the New Integral

Determine the y-limits by solving \(y = 2x\). When \(x = 0\), \(y = 0\). When \(x = 2\), \(y = 4\). Thus, for the new integral, y varies from \(x^2\) to \(2x\) for \(0 \leq x \leq 2\). Write the integral as \(\frac{y}{2} \leq x \leq 2\).

Key Concepts

Double Integrals

Double integrals are a way to integrate over a two-dimensional region. They extend the concept of a single integral to functions of two variables, like f(x, y). This allows us to calculate the volume under a surface or other properties like mass, depending on the function.

A double integral is written as follows: \ \ \ \[ \ \int\int_{R} f(x, y) \, dA \ \ \ \]
Here, \(R\) represents the region of integration in the xy-plane, and \(dA\) represents a small area element.

In our given integral, we have specific limits for x and y, which define this region.

Limits of Integration

The limits of integration are the boundaries within which we integrate. They tell us how x and y vary.
For our problem, the integral is given as:
\ \ \ \ \ \[ \int_{0}^{4} \ \int_{y/2}^{\sqrt{y}} f(x, y) dx dy \ \ \]

Here, the inner integral's limits are from \(\frac{y}{2}\) to \(\sqrt{y}\), and the outer integral's limits are from 0 to 4. These limits will help us sketch the region of integration and later switch the order of integration.

To switch the order of integration, we need to rewrite these limits for x first, then for the corresponding y values.

Region of Integration

The region of integration is the area on the xy-plane over which we integrate. To understand this region, we sketch the curves and find the intersection points.
For our problem, the curves are \(x = \frac{y}{2}\) and \(x = \sqrt{y}\).

Solving ......\...... for y, we find the intersection points at \(y = 0\) and \(y = 4\).

By drawing these curves, we locate the region of integration, which helps us define the new limits when reversing the integration order.
When switching, we observe x ranges from 0 to 2, and the corresponding y ranges can be found accordingly.

Sketching Curves

Sketching the curves helps us visualize the region of integration. Here's how we do it:

* Draw the curves \(x = \frac{y}{2}\) and \(x = \sqrt{y}\).

* Find their intersection points by equating \(\frac{y}{2}\) and \(\sqrt{y}\), giving us y = 0 and y = 4.

* Plot these points on the xy-plane.

* Identify the region bounded by these curves for 0 ≤ y ≤ 4.
* When reversing the integration order, observe how x ranges from 0 to 2 and find the corresponding y ranges.
* These visual aids ensure accurate integral limits, aiding in setting up the new integral correctly.